DYNAMICS OF A DAMPING OSCILLATOR WITH IMPACT AND IMPULSIVE EXCITATION

Tengfei Long; Guirong Jiang; Zhaosheng Feng

doi:10.11948/2015030

2015 Volume 5 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2015 > 5(3): 347-362

Next Article Previous Article

Tengfei Long, Guirong Jiang, Zhaosheng Feng. DYNAMICS OF A DAMPING OSCILLATOR WITH IMPACT AND IMPULSIVE EXCITATION[J]. Journal of Applied Analysis & Computation, 2015, 5(3): 347-362. doi: 10.11948/2015030

Citation:

Tengfei Long, Guirong Jiang, Zhaosheng Feng. DYNAMICS OF A DAMPING OSCILLATOR WITH IMPACT AND IMPULSIVE EXCITATION[J]. Journal of Applied Analysis & Computation, 2015, 5(3): 347-362. doi: 10.11948/2015030

DYNAMICS OF A DAMPING OSCILLATOR WITH IMPACT AND IMPULSIVE EXCITATION

1 School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China;
2 Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA

Fund Project:

Abstract

Abstract

There exist many types of external excitations that make the damping oscillator with impact have complex dynamics. In this study, both external impulsive excitation and impact are considered to construct a vibroimpact system. The fixed time pulse (impulsive excitation) and the state pulse (impact) lead to the complex and interesting dynamics. The conditions of the existence and stability of four kinds of periodic solutions are investigated, and the bifurcations of period-(1, 0) and period-(1, 1) solutions are analytically studied. Numerical simulations on periodic solutions and bifurcation diagrams are shown by the illustrative example.
- Damping oscillator /
- impulsive excitation /
- impact /
- periodic solution /
- bifurcation
MSC: 34C10;37C20;37M20

FullText(HTML)

References

[1]	A. Afsharfard and A. Farshidianfar, Free vibration analysis of nonlinear resilient impact dampers, Nonlinear Dyn., 73(2013), 155-166. Google Scholar
[2]	D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations:Periodic Solutions and Applications, Long man Scientific & Technical, New York, 1993. Google Scholar
[3]	P. Casini, O. Giannini and F. Vestroni, Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics, Nonlinear Dyn., 46(2006), 259-272. Google Scholar
[4]	S.N. Chow and S.W. Shaw, Bifurcations of subharmonics, J. Differential Equation, 65(1986), 304-320. Google Scholar
[5]	F.S. Collette, A combined tuned absorber and pendulum impact damper under random excitation, J. Franklin Institute, 216(1998), 199-213. Google Scholar
[6]	Z.D. Du and W.N. Zhang, Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. Math. Appl., 50(2005), 445-458. Google Scholar
[7]	W.C. Ding, G.F. Li, G.W. Luo and J.H. Xie, Torus T2 and its locking, doubling, chaos of a vibro-impact system, J. Franklin Institute, 349(2012), 337-348. Google Scholar
[8]	M.F. Dimentberg and D.V. Iourtchenko, Stochastic and/or chaotic response of a vibration system to imperfectly periodic sinusoidal excitation, Int. J. Bifurcation Chaos, 15(2005), 2057-2061. Google Scholar
[9]	Z. Feng, Duffing-van der Pol-type oscillator systems, Discrete Contin. Dyn. Syst. (Series S), 7(2014), 1231-1257. Google Scholar
[10]	Z. Feng and Q.G. Meng, First integrals for the damped Helmholtz oscillator, Int. J. Comput. Math., 87(2010), 2798-2810. Google Scholar
[11]	R. Giannini and R. Masiani, Non-Gaussian solution for random rocking of slender rigid block, Prob. Eng. Mech., 11(1996), 87-96. Google Scholar
[12]	C.B. Gan and H. Lei, Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations, J. Sound Vibration, 330(2011), 2174-2184. Google Scholar
[13]	P. Holmes, R.J. Full, D. Koditschek and J. Guckenheimer, Dynamics of legged locomotion:models, analysis and challenges, SIAM Rev., 48(2006), 207-304. Google Scholar
[14]	A. Ivanov, Bifurcations in impact systems, Chaos, Solitons & Fractals 7(1996), 1615-1634. Google Scholar
[15]	G.W. Luo and J.H. Xie, Hopf bifurcations and chaos of a two-degree-of-freedom vibro-impact system in two strong resonance cases, Int. J. Non-lin. Mech., 37(2002), 19-34. Google Scholar
[16]	A.C. Luo and L.D. Chen, Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos, Solitons & Fractals, 24(2005), 567-578. Google Scholar
[17]	S. Lenci and G. Rega, Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos Solitons Fractals, 11(2000), 2453-2472. Google Scholar
[18]	B.P. Mann, R.E. Carter and S.S. Hazra, Experimental study of an impact oscillator with viscoelastic and Hertzian contact, Nonlinear Dyn., 50(2007), 587-296. Google Scholar
[19]	S.Q. Ma, Q.S. Lu and Z. Feng, Double Hopf bifurcation for van der PolDuffing oscillator with parametric delay feedback control, J. Math. Anal. Appl., 338(2008), 993-1007. Google Scholar
[20]	A.B. Nordmark, Effects due to low velocity impact in mechanical oscillators, Int. J. Bifurcation Chaos, 2(1992), 597-605. Google Scholar
[21]	F. Peterka, Bifurcations and transition phenomena in an impact oscillator, Chaos, Solitons & Fractals, 7(1996), 1635-1647. Google Scholar
[22]	S.W. Shaw and R.H. Rand, The transition to chaos in a simple mechanical system, Int. J. Non-lin. Mech., 24(1989), 41-56. Google Scholar
[23]	J. Shen and Z.D. Du, Double impact periodic orbits for an inverted pendulum, Int. J. Non-lin. Mech., 46(2011), 1177-1190. Google Scholar
[24]	G.S. Whiston, An analytical model of two-dimensional impact/sliding response to harmonic excitation, J. Sound Vibration, 86(1983), 557-562. Google Scholar